Question

25x^4y^4-1
please explain step by step explanation
if u ans I will mark u as brainlist
​

Answer 1

Answer:

(5x²y²-1)(5x²y²+1)

Explanation :

25x⁴y⁴ - 1

= (5x²y²)² - 1²

_____________________

We know the algebraic identity:

a²-b² = (a-b)(a+b)

_____________________

= (5x²y²-1)(5x²y²+1)

••••

Answer 2

$25x^4y^4 - 1 \\ \\ \Rightarrow\ 25(xy)^4 - 1^2 \\ \\ \Rightarrow\ 25((xy)^2)^2-1^2 \\ \\ \Rightarrow\ 5^2 \times ((xy)^2)^2-1^2 \\ \\ \Rightarrow\ (5(xy)^2)^2 - 1^2 \ \ \ \ \ \ \ \ \ \ [a^2b^2 = (ab)^2] \\ \\ \Rightarrow\ (5(xy)^2+1)(5(xy)^2-1) \\ \\ \Rightarrow\ ((\sqrt{5})^2 \times (xy)^2+1)((\sqrt{5})^2 \times (xy)^2-1^2)$

$\Rightarrow\ ((xy\sqrt{5})^2+1)((xy\sqrt{5})^2-1^2) \\ \\ \Rightarrow\ ((xy\sqrt{5})^2+1)(xy \sqrt{5}+1)(xy \sqrt{5}-1)$

$\Rightarrow\ ((xy\sqrt{5})^2-(-1))(xy \sqrt{5}+1)(xy \sqrt{5}-1) \\ \\ \Rightarrow\ ((xy\sqrt{5})^2- \iota^2)(xy \sqrt{5}+1)(xy \sqrt{5}-1) \\ \\ \Rightarrow\ (xy\sqrt{5}+ \iota)(xy \sqrt{5}- \iota)(xy \sqrt{5}+1)(xy \sqrt{5}-1)$

OR

$25x^4y^4 - 1 \\ \\ \Rightarrow\ (xy \sqrt{5})^4-1^4 \\ \\ \Rightarrow\ ((xy \sqrt{5})^2+1^2)((xy \sqrt{5})^2-1^2)\\ \\ \Rightarrow\ ((xy \sqrt{5})^2+1)(xy \sqrt{5}+1)(xy \sqrt{5}-1) \\ \\ \Rightarrow\ ((xy \sqrt{5})^2-(-1))(xy \sqrt{5}+1)(xy \sqrt{5}-1) \\ \\ \Rightarrow\ ((xy\sqrt{5})^2- \iota^2)(xy \sqrt{5}+1)(xy \sqrt{5}-1) \\ \\ \Rightarrow\ (xy\sqrt{5}+ \iota)(xy \sqrt{5}- \iota)(xy \sqrt{5}+1)(xy \sqrt{5}-1)$

$[\iota=\sqrt{-1}]$

OR

Using the identity $a^4-b^4=(a+b)(a^3-a^2b+ab^2-b^3)$,

$25x^4y^4 - 1 \\ \\ \Rightarrow\ (xy \sqrt{5})^4-1^4 \\ \\ \Rightarrow\ (xy \sqrt{5}+1)((xy \sqrt{5})^3-(xy \sqrt{5})^2 \times 1 + (xy \sqrt{5}) \times 1^2 -1^3) \\ \\ \Rightarrow\ (xy \sqrt{5}+1)((xy \sqrt{5})^3-(xy \sqrt{5})^2 + xy \sqrt{5} -1) \\ \\ \Rightarrow\ (xy \sqrt{5}+1)((xy \sqrt{5})^3-(xy \sqrt{5})^2 + (xy \sqrt{5})^1 - (xy \sqrt{5})^0)$

OR

Using the identity $a^4-b^4=(a-b)(a^3+a^2b+ab^2+b^3)$,

$25x^4y^4 - 1 \\ \\ \Rightarrow\ (xy \sqrt{5})^4-1^4 \\ \\ \Rightarrow\ (xy \sqrt{5}-1)((xy \sqrt{5})^3+(xy \sqrt{5})^2 \times 1 + (xy \sqrt{5}) \times 1^2 +1^3) \\ \\ \Rightarrow\ (xy \sqrt{5}-1)((xy \sqrt{5})^3+(xy \sqrt{5})^2 + xy \sqrt{5} +1) \\ \\ \Rightarrow\ (xy \sqrt{5}-1)((xy \sqrt{5})^3+(xy \sqrt{5})^2 + (xy \sqrt{5})^1+(xy \sqrt{5})^0)$

Hope these help you.

Plz mark it as the brainliest if they help.

Plz ask me if you've any doubts on my answer.

Thank you. :-))